1. New Adaptive Localization Algorithms That Achieve Better Coverage for Wireless Sensor Networks
Advisor: Chiuyuan Chen
Student: Shao-Chun Lin
Department of Applied Mathematics National Chiao Tung University
2013/8/11

2. Introduction

3. Sensor能夠彼此間在傳送半徑內傳送訊息
藉由接力的方式將訊息傳到一個主控台
使用者能夠獲得這些資訊來判斷到底這個區域發生什麼事情
圖片來源:

4. transmission range (𝑅)
Node : Sensor
Disk radius:
transmission range (𝑅) 𝑅

6. Unit Disk Graph
Node : Sensor
Disk radius: Transmission range (𝑅)

7. Applications of wireless sensor networks (WSNs)
Wildlife tracking, military, forest fire detection,
temperature detection,
environment monitoring
Why localization?
To detect and record events.
When tracking objects, the position information
is important. …

8. Related works and Main Results

9. Definitions initial-anchor a node equipped with GPS
initial-anchor set (𝑺)
the set containing all initial-anchors
anchor
a node knows its position.
feasible
The initial-anchor set 𝑆 is called feasible
if the position of each node in the given graph
can be determined with 𝑆.
為了要解答這個問題，我們需要下面幾個定義 以同一個圖來說，不同的定位演算法會有不同的initial anchor set

10. Informations can be used to localize 𝑣
For each node 𝑣, the distances between 𝑢,𝑣 where 𝑢∈𝑁(𝑣)
The positions of anchors in 𝑁 𝑣

11. *Fine-grain Localization Coarse-grain Localization
Localization types
*Fine-grain Localization
Coarse-grain Localization
Resitrict 𝑆 =3 [Rigid Theory]
Consider noise [11~14, 2001~]
*Find a feasible 𝑆 with |𝑆| as small as possible [8, 2011 Huang]
Best Coverage [11~14, 2001~]
圖片來源:Efficient Location Training Protocols for Heterogeneous Sensor and Actor Networks

12. *Fine-grain Localization Coarse-grain Localization
Localization types
*Fine-grain Localization
Coarse-grain Localization
Resitrict 𝑆 =3 [Rigid Theory]
Consider noise [11~14, 2001~]
*Find a feasible 𝑆 with |𝑆| as small as possible [8, 2011 Huang]
Best Coverage [11~14, 2001~]
圖片來源:Efficient Location Training Protocols for Heterogeneous Sensor and Actor Networks

13. Rigidity Theory non-rigid : localization solution is infinite.
rigid : localization solution is finite.
globally rigid : localization solution is unique.

14. Non-rigid
Infinite
Initial-anchor
Unknown

15. Rigid graph
Finite

16. Globally rigid graph
Unique

17. Characterize globally rigid graph
A graph which exists 3 anchors has unique localization solution if and only if the graph is globally rigid.
redundantly rigid: After one edge is deleted, the remaining graph is a rigid graph.
Laman’s Condition ([2], 1970 Laman) A graph 𝐺=(𝑉,𝐿) with 𝑛 vertices is rigid in 𝑅 2 if and only if 𝐿 contains a subset 𝐸 consisting of 𝟐𝒏 − 𝟑 edges with the property that, for any nonempty subset 𝐸′⊂ 𝐸, the number of edges in 𝐸′ cannot exceed 𝟐𝒏′ − 𝟑, where 𝑛’ is the number of vertices of 𝐺 which are endpoints of edges in 𝐸′.

18. Characterize globally rigid graph
1982, Lovasz and Yemini shows 6-connected graph is redundantly rigid.
[7] 1992, Hendrickson proposed a polynomial-time algorithm to determine the redundantly rigidity of a graph.
Hendrickson’s Conjecture
A graph is called globally rigid if and only if the graph is 3-connected and redundantly rigid.
[9] 2005, Jackson et al. proved that Hendrickson’s Conjecture is true.
[5] 2005, Connelly mentioned that there is an algorithm to determine if a graph is globally rigid (i.e. localizable) in polynomial-time.
C-algorithm.

19. Characterize globally rigid graph
C-algorithm cannot compute position.
2006, Aspnes shows that to compute position in globally rigid with 3 anchors is NP-hard

20. *Fine-grain Localization Coarse-grain Localization
Localization types
*Fine-grain Localization
Coarse-grain Localization
Resitrict 𝑆 =3 [Rigid Theory]
Consider noise [11~14, 2001~]
*Find a feasible 𝑆 with |𝑆| as small as possible [8, 2011 Huang]
Best Coverage [11~14, 2001~]
圖片來源:Efficient Location Training Protocols for Heterogeneous Sensor and Actor Networks

21. Choose node to become initial-anchor
Grounded, generic, UDG
A graph G
Choose node to become initial-anchor
Nodes with degree ≤2
Trilateration
*Tri + Sweep2
Localization-Phase
AnchorChoose-Phase
*Tri + Rigid
Check if all nodes are localized No
Yes
HuangChoose[2011]
Output a feasible initial-anchor set 𝑆
*MaxDegree Choose
*AdaptiveChoose

22. Choose node to become initial-anchor
Grounded, generic, UDG
A graph G
Choose node to become initial-anchor
Nodes with degree ≤2
Trilateration
*Tri + Sweep2
Localization-Phase
AnchorChoose-Phase
*Tri + Rigid
Check if all nodes are localized No
Yes
HuangChoose[2011]
Output a feasible initial-anchor set 𝑆
*MaxDegree Choose
*AdaptiveChoose

23. The graph we considered in this thesis
Unit Disk Graph
grounded ([2], 2005 Aspnes et al.) A graph 𝐺=(𝑉,𝐸) is grounded if 𝑢𝑣∈𝐸 implies that the distance 𝑢𝑣 can be measured or estimated via wireless communication.
generic
A graph is called generic if node coordinates are algebraically independent over rationals.

24. Choose node to become initial-anchor
A graph G
Choose node to become initial-anchor
Nodes with degree ≤2
Localization-Phase
AnchorChoose-Phase
Check if all nodes are localized No
Yes
Output a feasible initial-anchor set 𝑆

25. Theorem:
Let 𝑆 be any feasible initial-anchor set of 𝐺. For all 𝑣∈ 𝑉 with degree deg 𝑣 ≤2, we have 𝑣∈𝑆.

26. Choose node to become initial-anchor
A graph G
Choose node to become initial-anchor
Trilateration
*Tri + Sweep2
Localization-Phase
AnchorChoose-Phase
*Tri + Rigid
Check if all nodes are localized No
Yes
Output a feasible initial-anchor set 𝑆

27. Localization-Phase Trilateration Sweep2+Tri Rigid+Tri anchor unknown
initial-anchor

28. Trilateration
anchor
unknown
initial-anchor

29. Trilateration
anchor
unknown
initial-anchor

30. Sweep2(+Tri) 𝑢 𝑣

31. Sweep2(+Tri) 𝑢 𝑣

32. Sweep2
2006 Goldenberg first propose this idea, and called this as sweep.
[8] 2011, Huang modified it to 2 neighbors version by two cases.
In 2013, this thesis simplifies it and achieves the same performance, called this algorithm as Sweep2.

33. Rigidity Theory A point formation is a set of node positions.
A point formation is called a localization solution of a grounded graph 𝐺=(𝑉,𝐸) if the following condition holds: for each 𝑢𝑣∈𝐸, distance between 𝑢, 𝑣 calculated from the their positions equals the distance 𝑢𝑣 in grounded graph 𝐺.
A framework is that for each node v, v has its position. Framework:每個節點的位置
A framework is called as localization solution if the following condition holds;for each pair nodes u,v in G, if uv in G then uv distance satisfy the distance in grounded graph.

34. Rigid(+Tri)
Localized subgraph
Subgraph 𝑢

35. Rigid(+Tri)
Localized subgraph
Subgraph 𝑢

36. Rigid(Tri)
Localized subgraph
Subgraph 𝑢

37. Choose node to become initial-anchor
A graph G
Choose node to become initial-anchor
Localization-Phase
AnchorChoose-Phase
Check if all nodes are localized No
Yes
HuangChoose[2011]
Output a feasible initial-anchor set 𝑆
*MaxDegree Choose
*AdaptiveChoose

38. AnchorChoose-Phase 𝒗.ann : # of anchors in 𝑁(𝑣)
MaxDegreeChoose (a straightforward approach)
HuangChoose ([8] 2011, Huang et al.) 𝒗.𝒓𝒂𝒏𝒌𝟎=# of 𝑢∈𝑁 𝑣 with 𝑢.ann =0 𝒗.𝒓𝒂𝒏𝒌𝟏=# of 𝑢∈𝑁 𝑣 with 𝑢.ann=1
𝒗.𝒓𝒂𝒏𝒌𝟐=# of 𝑢∈𝑁 𝑣 with 𝑢.ann=2
Choose 𝒗 with maximum 𝒗.𝒓𝒂𝒏𝒌𝟐 -> 𝒗.𝒓𝒂𝒏𝒌1 -> 𝒗.𝒓𝒂𝒏𝒌0
AdaptiveChoose (This thesis)
Choose 𝒗 with maximum 𝒅𝒆𝒈 𝒗 −𝒗.ann

39. Choose node to become initial-anchor
A graph G
Choose node to become initial-anchor
Localization-Phase
AnchorChoose-Phase
Check if all nodes are localized No
Yes
Output a feasible initial-anchor set 𝑆

40. 𝑆 ≤𝑘 A graph G Choose node to become initial-anchor Localization-Phase
AnchorChoose-Phase
Check if all nodes are localized or 𝑆 ≥𝑘 No
Yes
𝑨𝒏𝒄𝒉𝒐𝒓𝒔: The set of nodes that know their positions
Output an initial-anchor set 𝑆 and |𝐴𝑛𝑐ℎ𝑜𝑟𝑠|

41. Simulation

42. Simulation Localization-Phase AnchorChoose-Phase
Trilateration (LocalTri)
Sweep2
AnchorChoose-Phase
HuangChoose ([8] 2005, Huang et al.)
AdaptiveChoose
MaxDegreeChoose (MaxDegree)

43. Simulations Notation IAF: cardinality of an initial-anchor set
𝐴: Algorithm
𝐴𝑛𝑐ℎ𝑜𝑟𝑠: The set of nodes that know their positions
𝑆: initial-anchor set
𝑛: # of nodes
IAF: cardinality of an initial-anchor set
𝐼𝐴 𝐹 𝐴 𝐺 =|𝑆|
COVERAGE: the percentage of nodes that know their positions,
𝐶𝑂𝑉𝐸𝑅𝐴𝐺𝐸 𝐴 𝐺 = |𝐴𝑛𝑐ℎ𝑜𝑟𝑠| 𝑛

44. Sparse* 200 nodes in 800m × 600m R=70, k=30 and 50 5~7 2.89%
Average Degree
𝐸 |𝑉| 2
G 200 nodes in 1200m × 1000m R=80
6.56
3.30%
Vary Sparse*
200 nodes in 1200m × 1000m R=70, k=100 and 120
2~3
1.19%
Sparse* 200 nodes in 800m × 600m R=70, k=30 and 50
5~7
2.89%
Dense* 200 nodes in 800m × 600m R=100, k=5 and 10
10~14
5.71%
*200 graphs for each

45. G
圖片來源:Minimum cost localization problem in wireless sensor networks

46. *𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐻𝑢𝑛𝑎𝑔𝐻ℎ𝑜𝑜𝑠𝑒 𝐺 =33 𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =31G
𝐼𝐴 𝐹 𝐴 𝐺 =|𝑆|
*𝐼𝐴 𝐹 𝑇𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛+𝐻𝑢𝑎𝑛𝑔𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =42 𝐼𝐴 𝐹 𝑇𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛+𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =40
*𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐻𝑢𝑛𝑎𝑔𝐻ℎ𝑜𝑜𝑠𝑒 𝐺 =33 𝐼𝐴 𝐹 𝑆𝑤𝑒𝑒𝑝2+𝐴𝑑𝑎𝑝𝑡𝑖𝑣𝑒𝐶ℎ𝑜𝑜𝑠𝑒 𝐺 =31
*Referred to [8] 2011, Huang et al.

47. Very Sparse Graphs IAF: cardinality of an initial-anchor set
Average Blue: Red: Green:156
IAF: cardinality of an initial-anchor set

48. Very Sparse Graphs IAF: cardinality of an initial-anchor set
Average Blue: Red: Green:
IAF: cardinality of an initial-anchor set

49. Sparse Graphs IAF: cardinality of an initial-anchor set
Average Blue:62.26 Red: Green:
IAF: cardinality of an initial-anchor set

50. Sparse Graphs IAF: cardinality of an initial-anchor set
Average Blue: Red:50.36
Green:51.005
IAF: cardinality of an initial-anchor set

51. Dense Graphs IAF: cardinality of an initial-anchor set
Average Blue:11.72 Red:6.915 Green:6.875
IAF: cardinality of an initial-anchor set

52. Dense Graphs IAF: cardinality of an initial-anchor set
Average Blue:6.79 Red:5.81 Green:5.795
IAF: cardinality of an initial-anchor set

53. Very Sparse Graphs
Average Blue:57.73%
Red:57.80% Green:57.80%
𝑘=100

54. Very Sparse Graphs
Average Blue:57.92% Red:57.87% Green:57.87%
𝑘=120

55. Very Sparse Graphs
Average Blue:65.64% Red:68.06% Green:68.10%
𝑘=100

56. Very Sparse Graphs
Average Blue:71.44% Red:71.37% Green:71.43%
𝑘=120

57. Sparse Graphs
Average Blue:24.26%
Red:48.66% Green:48.75%
𝑘=30

58. Sparse Graphs
Average Blue:76.64% Red:88.58% Green:88.70%
𝑘=50

59. Sparse Graphs
Average Blue:49.69%
Red:60.23% Green:60.47%
𝑘=30

60. Sparse Graphs
Average Blue:97.08% Red:97.78% Green:97.58%
𝑘=50

61. Dense Graphs
Average Blue:12.19% Red:80.49% Green:82.09%
𝑘=5

62. Dense Graphs
Average Blue:70.36% Red:99.92% Green:99.88%
𝑘=10

63. Dense Graphs
Average Blue:68.59% Red:85.50% Green:87.31%
𝑘=5

64. Dense Graphs
Average Blue:99.69% Red:99.99% Green:99.97%
𝑘=10

65. A=LocalTri 𝐺 No data Vary Sparse The same Sparse Dense AdaptiveChoose
IAF
COVERAGE 𝐺
AdaptiveChoose
No data
Vary Sparse
HuangChoose ≈AdaptiveChoose
The same
Sparse
MaxDegree ≈AdaptiveChoose
Dense

66. 𝐺 No data Vary Sparse The same Sparse Dense A=Sweep2 AdaptiveChoose
IAF
COVERAGE 𝐺
AdaptiveChoose
No data
Vary Sparse
HuangChoose ≈AdaptiveChoose
The same
Sparse
≈MaxDegree
Dense
MaxDegree ≈AdaptiveChoose

67. Concluding remarks
Sweep2 are simpler than Sweep ([8] 2005, Huang) but cover all the cases.
A new algorithm for rigid in Localization-Phase

68. Future works A much powerful Greedy algorithms to choose anchors.
Combine AdpativeChoose and HuangChoose to obtain better result.
Given a certain initial-anchor set, determine what kind of graphs are localizable.
Design a distributed version of AdaptiveChoose.

69. Thank you for your attention!